Search: a097136 -id:a097136
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A097135
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a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).
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+10
5
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1, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f. : (1+2*x-x^2)/(1-x-x^2).
a(n) = a(n-1)+a(n-2) for n>2.
a(2n) = A097134(n); a(2n+1) = 3*F(2n+1).
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A097133
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a(n) = 3*Fibonacci(n) + (-1)^n.
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+10
4
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1, 2, 4, 5, 10, 14, 25, 38, 64, 101, 166, 266, 433, 698, 1132, 1829, 2962, 4790, 7753, 12542, 20296, 32837, 53134, 85970, 139105, 225074, 364180, 589253, 953434, 1542686, 2496121, 4038806, 6534928, 10573733, 17108662, 27682394, 44791057, 72473450, 117264508
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1+2*x+2*x^2)/((1+x)*(1-x-x^2));
a(n) = 2*a(n-2)+a(n-3);
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MATHEMATICA
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CoefficientList[Series[(1+2x+2x^2)/((1+x)(1-x-x^2)), {x, 0, 40}], x] (* or *) LinearRecurrence[{0, 2, 1}, {1, 2, 4}, 40] (* Harvey P. Dale, May 07 2011 *)
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PROG
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(Haskell)
a097133 n = a097133_list !! n
a097133_list = 1 : 2 : 4 : zipWith (+)
(map (* 2) $ tail a097133_list) a097133_list
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A192908
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Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
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+10
3
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1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199
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OFFSET
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0,3
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COMMENTS
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The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.
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LINKS
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FORMULA
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a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
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MATHEMATICA
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u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a; p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4, -4, 1}, {1, 1, 3, 7}, 30] (* G. C. Greubel, Jan 11 2019 *)
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PROG
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(PARI) vector(30, n, n--; if(n==0, 1, 1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019
(Magma) [1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019
(Sage) [1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019
(GAP) Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A097132
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a(n) = Sum_{k=0..n} Fibonacci(k) + (-1)^k*Fibonacci(k-1).
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+10
1
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1, 2, 4, 5, 10, 12, 25, 30, 64, 77, 166, 200, 433, 522, 1132, 1365, 2962, 3572, 7753, 9350, 20296, 24477, 53134, 64080, 139105, 167762, 364180, 439205, 953434, 1149852, 2496121, 3010350, 6534928, 7881197, 17108662, 20633240, 44791057
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graph;
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1 + x - x^2 - 2*x^3)/((1 - 3*x^2 + x^4)*(1-x));
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - a(n-4) + a(n-5);
a(n) = 1 + (1/2 - sqrt(5)/2)^n*(1/2 - 3*sqrt(5)/10) - (sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 + 1/2) + (-sqrt(5)/2 - 1/2)^n*(3*sqrt(5)/10 - 1/2) + (sqrt(5)/2 + 1/2)^n*(3*sqrt(5)/10 + 1/2);
a(2n) = 1 + 3*Fibonacci(2n) = A097136(n);
a(2n+1) = 1 + Fibonacci(2n) + Fibonacci(2n+2) = 1 + Lucas(2n).
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MATHEMATICA
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LinearRecurrence[{1, 3, -3, -1, 1}, {1, 2, 4, 5, 10}, 40] (* Harvey P. Dale, Nov 12 2022 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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